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arxiv: 2511.19204 · v3 · submitted 2025-11-24 · 💻 cs.RO · cs.SY· eess.SY

Reference-Free Sampling-Based Model Predictive Control

Fabian Schramm , Pierre Fabre , Nicolas Perrin-Gilbert , Justin Carpentier This is my paper

Pith reviewed 2026-05-17 06:14 UTC · model grok-4.3

classification 💻 cs.RO cs.SYeess.SY
keywords sampling-based MPCmodel predictive path integralquadrupedal locomotionemergent gaitscubic Hermite splinesreference-free controlreal-time control
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The pith

Sampling-based MPC with cubic Hermite splines discovers emergent gaits and jumps on quadruped robots without references or pre-training.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper presents a sampling-based model predictive control framework that parameterizes trajectories with cubic Hermite splines to optimize high-level objectives. This setup lets robots automatically discover diverse motion patterns such as trotting, galloping, jumping, standing, and handstand balancing by adapting contact-making and contact-breaking strategies on the fly. The approach requires only a modest number of sampled trajectories, which enables real-time execution on standard CPU hardware rather than needing GPU acceleration. A sympathetic reader would care because it removes reliance on handcrafted gait patterns, predefined contact sequences, reference tracking, and offline pre-training, potentially allowing more flexible robot behaviors to emerge directly from task goals.

Core claim

The authors claim that integrating a cubic Hermite spline parameterization of position and velocity control points into a model predictive path integral sampling framework enables the discovery of diverse motion patterns ranging from trotting to galloping, robust standing policies, jumping, and handstand balancing purely through the optimization of high-level objectives. This works on the Go2 quadrupedal robot and, in simulation, on a Humanoid, all without requiring reference tracking or offline pre-training, while maintaining sample efficiency for real-time CPU control.

What carries the argument

Cubic Hermite spline parameterization of position and velocity control points within the model predictive path integral (MPPI) sampling framework, which carries the argument by enabling automatic adaptation of contact strategies with few samples.

If this is right

  • The method generates trotting, galloping, robust standing, jumping, and handstand balancing on the Go2 quadruped.
  • In simulation it produces backflips, dynamic handstand balancing, and locomotion on a Humanoid.
  • Real-time control runs on standard CPU hardware using only a limited number of sampled trajectories.
  • All behaviors emerge without reference tracking or offline pre-training.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • This sampling efficiency could reduce manual engineering effort when deploying controllers on new robot platforms.
  • The automatic contact adaptation might extend to tasks involving variable terrain or external disturbances.
  • Combining the spline parameterization with learned dynamics models could improve performance under model mismatch.

Load-bearing premise

The underlying dynamics model is accurate enough that sampling a modest number of spline-parameterized trajectories will reliably discover effective contact sequences and gaits without reference tracking or offline pre-training.

What would settle it

Running the method on the physical Go2 robot and observing whether it produces unstable gaits or requires far more samples when the dynamics model contains moderate errors such as unmodeled friction changes or actuator delays.

Figures

Figures reproduced from arXiv: 2511.19204 by Fabian Schramm, Justin Carpentier, Nicolas Perrin-Gilbert, Pierre Fabre.

Figure 1
Figure 1. Figure 1: Overview of the reference-free sampling-based MPC framework (top): our approach enables emergent jumping motion experimentally achieved on the Go2 robot without any guiding reference (bottom). These methods also necessitate hand-crafted cost functions to obtain good contact sequences [11]. Sampling-based methods offer an attractive alternative by providing derivative-free optimization that is inherently we… view at source ↗
Figure 2
Figure 2. Figure 2: Sequence illustrating the discovered walking gait on the Go2 quadruped. [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: Plot comparison of different spline types with the same interpolation points (red), resulting in different normalized position and velocity trajectories. Cubic Hermite splines exhibit a lower variance than quadratic and cubic splines, resulting in finer sampling granularity. frequently overshoot these limits, our cubic Hermite for￾mulation respects the bounds throughout the trajectory. It is worth noting t… view at source ↗
Figure 4
Figure 4. Figure 4: The nominal trajectory (black) evolves through spline [PITH_FULL_IMAGE:figures/full_fig_p004_4.png] view at source ↗
Figure 5
Figure 5. Figure 5: Smooth transitioning from trotting to galloping as [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: Robot base height during vertical jumping. When [PITH_FULL_IMAGE:figures/full_fig_p006_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: Base pitch trajectory during handstand pose. [PITH_FULL_IMAGE:figures/full_fig_p007_7.png] view at source ↗
Figure 8
Figure 8. Figure 8: Emergent dynamic behaviors in simulation. 5) Humanoid locomotion: To highlight the generality of our reference-free framework beyond quadrupedal locomo￾tion, we also evaluate it on the G1 humanoid in simulation. Despite the shift in morphology and increase in DoFs (from 12 to 37), the same algorithm successfully discovers a walk￾ing gait from the exact high-level cost without modification, as displayed wit… view at source ↗
read the original abstract

We present a sampling-based model predictive control (MPC) framework that enables emergent locomotion without relying on handcrafted gait patterns or predefined contact sequences. Our method discovers diverse motion patterns, ranging from trotting to galloping, robust standing policies, jumping, and handstand balancing, purely through the optimization of high-level objectives. Building on model predictive path integral (MPPI), we propose a cubic Hermite spline parameterization that operates on position and velocity control points. Our approach enables contact-making and contact-breaking strategies that adapt automatically to task requirements, requiring only a limited number of sampled trajectories. This sample efficiency enables real-time control on standard CPU hardware, eliminating the GPU acceleration typically required by other state-of-the-art MPPI methods. We validate our approach on the Go2 quadrupedal robot, demonstrating a range of emergent gaits and basic jumping capabilities. In simulation, we further showcase more complex behaviors, such as backflips, dynamic handstand balancing and locomotion on a Humanoid, all without requiring reference tracking or offline pre-training.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 2 minor

Summary. The manuscript presents a sampling-based MPC framework extending MPPI with cubic Hermite spline parameterization over position and velocity control points. It claims to discover emergent, reference-free locomotion behaviors (trotting, galloping, jumping, handstand balancing) on quadrupeds and humanoids solely by optimizing high-level objectives, with hardware validation on the Unitree Go2 and simulation results, while achieving real-time performance on standard CPU hardware without GPU acceleration.

Significance. If the central claims are supported by quantitative evidence, the work would be significant for legged-robot control: it removes the need for handcrafted gait patterns, contact schedules, or offline pre-training, and the reported CPU efficiency could broaden deployment of sampling-based methods on resource-limited platforms.

major comments (2)
  1. [Experiments] Experiments section: no quantitative metrics (success rates, rollout times, cost values), baselines, or ablations are reported for the Go2 hardware trials or the humanoid simulation tasks. Without these data it is impossible to evaluate the claimed sample efficiency or the assertion that a modest number of spline-parameterized trajectories reliably discovers contact sequences.
  2. [Method] Method and Dynamics sections: the central claim that high-level objectives alone suffice to surface stable contact-making/breaking sequences rests on the untested assumption that the forward dynamics model accurately captures friction and contact events. Model mismatch (explicitly flagged as a risk for sim-to-real transfer) could cause sampled trajectories to violate feasibility or converge to unstable policies; no sensitivity analysis or hardware-model discrepancy quantification is provided.
minor comments (2)
  1. [Abstract] Abstract: the phrase 'a range of emergent gaits' is vague; specify which behaviors were demonstrated on hardware versus simulation.
  2. [Method] Notation: clarify whether the spline control points are optimized directly or via the MPPI importance-sampling weights; the current description leaves the exact mapping between spline parameters and the MPPI noise distribution ambiguous.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive feedback. We address each major comment below and indicate the revisions made to strengthen the manuscript.

read point-by-point responses

  1. Referee: [Experiments] Experiments section: no quantitative metrics (success rates, rollout times, cost values), baselines, or ablations are reported for the Go2 hardware trials or the humanoid simulation tasks. Without these data it is impossible to evaluate the claimed sample efficiency or the assertion that a modest number of spline-parameterized trajectories reliably discovers contact sequences.

    Authors: We agree that the original experiments section would benefit from explicit quantitative metrics and comparisons. In the revised manuscript we have added success rates for each discovered behavior on the Go2, average cost values and rollout times for both hardware and humanoid simulation trials, a baseline comparison against standard MPPI, and an ablation study varying the number of sampled trajectories to quantify sample efficiency. revision: yes

  2. Referee: [Method] Method and Dynamics sections: the central claim that high-level objectives alone suffice to surface stable contact-making/breaking sequences rests on the untested assumption that the forward dynamics model accurately captures friction and contact events. Model mismatch (explicitly flagged as a risk for sim-to-real transfer) could cause sampled trajectories to violate feasibility or converge to unstable policies; no sensitivity analysis or hardware-model discrepancy quantification is provided.

    Authors: We acknowledge that a dedicated sensitivity analysis and explicit quantification of hardware-model discrepancy were not included in the initial submission. The successful real-world transfer on the Unitree Go2 provides supporting evidence that the model is sufficiently accurate for the observed behaviors; however, to directly address the concern we have added a new analysis section that reports trajectory discrepancies between simulation and hardware for representative gaits and includes a sensitivity study to friction coefficient variations. revision: yes

Circularity Check

0 steps flagged

No circularity: method extends independent MPPI framework via new spline parameterization

full rationale

The paper's derivation chain starts from the established MPPI sampling procedure and adds a cubic Hermite spline representation over position/velocity control points. All subsequent claims about emergent contact-making/breaking, gait discovery, and real-time CPU performance follow directly from rolling out the sampled trajectories under the given dynamics model and optimizing the high-level cost; none of these quantities are defined in terms of themselves or obtained by fitting parameters to the target behaviors. No self-citation is invoked as a load-bearing uniqueness theorem, and the spline choice is presented as an explicit design decision rather than smuggled via prior work. The derivation therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 0 axioms · 0 invented entities

The approach inherits standard assumptions of model-based predictive control and the MPPI sampling procedure; no new free parameters or invented entities are introduced in the abstract.

pith-pipeline@v0.9.0 · 5484 in / 999 out tokens · 39269 ms · 2026-05-17T06:14:48.427743+00:00 · methodology

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Forward citations

Cited by 1 Pith paper

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score.

  1. Sampling-Based Control via Entropy-Regularized Optimal Transport

    cs.RO 2026-05 unverdicted novelty 7.0

    OT-MPC computes an optimal coupling between candidate control sequences and low-cost proposals via entropy-regularized optimal transport and the Sinkhorn algorithm to improve sampling-based MPC performance.

Reference graph

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